\(\int (e x)^m \tanh ^2(a+2 \log (x)) \, dx\) [171]

Optimal result

Integrand size = 15, antiderivative size = 79 \[ \int (e x)^m \tanh ^2(a+2 \log (x)) \, dx=\frac {(e x)^{1+m}}{e (1+m)}+\frac {(e x)^{1+m}}{e \left (1+e^{2 a} x^4\right )}-\frac {(e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{4},\frac {5+m}{4},-e^{2 a} x^4\right )}{e} \] Output:

(e*x)^(1+m)/e/(1+m)+(e*x)^(1+m)/e/(1+exp(2*a)*x^4)-(e*x)^(1+m)*hypergeom([ 
1, 1/4+1/4*m],[5/4+1/4*m],-exp(2*a)*x^4)/e
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00 \[ \int (e x)^m \tanh ^2(a+2 \log (x)) \, dx=-\frac {x (e x)^m \left (-1+4 \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{4},\frac {5+m}{4},-x^4 (\cosh (2 a)+\sinh (2 a))\right )-4 \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{4},\frac {5+m}{4},-x^4 (\cosh (2 a)+\sinh (2 a))\right )\right )}{1+m} \] Input:

Integrate[(e*x)^m*Tanh[a + 2*Log[x]]^2,x]
 

Output:

-((x*(e*x)^m*(-1 + 4*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, -(x^4*(Cos 
h[2*a] + Sinh[2*a]))] - 4*Hypergeometric2F1[2, (1 + m)/4, (5 + m)/4, -(x^4 
*(Cosh[2*a] + Sinh[2*a]))]))/(1 + m))
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.23, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6071, 963, 27, 959, 888}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (e x)^m \tanh ^2(a+2 \log (x)) \, dx\)

\(\Big \downarrow \) 6071

\(\displaystyle \int \frac {\left (e^{2 a} x^4-1\right )^2 (e x)^m}{\left (e^{2 a} x^4+1\right )^2}dx\)

\(\Big \downarrow \) 963

\(\displaystyle \frac {(e x)^{m+1}}{e \left (e^{2 a} x^4+1\right )}-\frac {1}{4} e^{-4 a} \int \frac {4 (e x)^m \left (e^{4 a} m-e^{6 a} x^4\right )}{e^{2 a} x^4+1}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(e x)^{m+1}}{e \left (e^{2 a} x^4+1\right )}-e^{-4 a} \int \frac {(e x)^m \left (e^{4 a} m-e^{6 a} x^4\right )}{e^{2 a} x^4+1}dx\)

\(\Big \downarrow \) 959

\(\displaystyle \frac {(e x)^{m+1}}{e \left (e^{2 a} x^4+1\right )}-e^{-4 a} \left (e^{4 a} (m+1) \int \frac {(e x)^m}{e^{2 a} x^4+1}dx-\frac {e^{4 a} (e x)^{m+1}}{e (m+1)}\right )\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {(e x)^{m+1}}{e \left (e^{2 a} x^4+1\right )}-e^{-4 a} \left (\frac {e^{4 a} (e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{4},\frac {m+5}{4},-e^{2 a} x^4\right )}{e}-\frac {e^{4 a} (e x)^{m+1}}{e (m+1)}\right )\)

Input:

Int[(e*x)^m*Tanh[a + 2*Log[x]]^2,x]
 

Output:

(e*x)^(1 + m)/(e*(1 + E^(2*a)*x^4)) - (-((E^(4*a)*(e*x)^(1 + m))/(e*(1 + m 
))) + (E^(4*a)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, -( 
E^(2*a)*x^4)])/e)/E^(4*a)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 959

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p 
+ 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p 
 + 1) + 1))   Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, 
 n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
 

rule 963

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^2, x_Symbol] :> Simp[(-(b*c - a*d)^2)*(e*x)^(m + 1)*((a + b*x^n)^(p + 1) 
/(a*b^2*e*n*(p + 1))), x] + Simp[1/(a*b^2*n*(p + 1))   Int[(e*x)^m*(a + b*x 
^n)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + b^2*c^2*n*(p + 1) + a*b*d^2*n*(p + 
 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] 
 && IGtQ[n, 0] && LtQ[p, -1]
 

rule 6071

Int[((e_.)*(x_))^(m_.)*Tanh[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] 
 :> Int[(e*x)^m*((-1 + E^(2*a*d)*x^(2*b*d))^p/(1 + E^(2*a*d)*x^(2*b*d))^p), 
 x] /; FreeQ[{a, b, d, e, m, p}, x]
 

Maple [F]

\[\int \left (e x \right )^{m} \tanh \left (a +2 \ln \left (x \right )\right )^{2}d x\]

Input:

int((e*x)^m*tanh(a+2*ln(x))^2,x)
 

Output:

int((e*x)^m*tanh(a+2*ln(x))^2,x)
 

Fricas [F]

\[ \int (e x)^m \tanh ^2(a+2 \log (x)) \, dx=\int { \left (e x\right )^{m} \tanh \left (a + 2 \, \log \left (x\right )\right )^{2} \,d x } \] Input:

integrate((e*x)^m*tanh(a+2*log(x))^2,x, algorithm="fricas")
 

Output:

integral((e*x)^m*tanh(a + 2*log(x))^2, x)
 

Sympy [F]

\[ \int (e x)^m \tanh ^2(a+2 \log (x)) \, dx=\int \left (e x\right )^{m} \tanh ^{2}{\left (a + 2 \log {\left (x \right )} \right )}\, dx \] Input:

integrate((e*x)**m*tanh(a+2*ln(x))**2,x)
 

Output:

Integral((e*x)**m*tanh(a + 2*log(x))**2, x)
 

Maxima [F]

\[ \int (e x)^m \tanh ^2(a+2 \log (x)) \, dx=\int { \left (e x\right )^{m} \tanh \left (a + 2 \, \log \left (x\right )\right )^{2} \,d x } \] Input:

integrate((e*x)^m*tanh(a+2*log(x))^2,x, algorithm="maxima")
 

Output:

integrate((e*x)^m*tanh(a + 2*log(x))^2, x)
 

Giac [F]

\[ \int (e x)^m \tanh ^2(a+2 \log (x)) \, dx=\int { \left (e x\right )^{m} \tanh \left (a + 2 \, \log \left (x\right )\right )^{2} \,d x } \] Input:

integrate((e*x)^m*tanh(a+2*log(x))^2,x, algorithm="giac")
 

Output:

integrate((e*x)^m*tanh(a + 2*log(x))^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^m \tanh ^2(a+2 \log (x)) \, dx=\int {\mathrm {tanh}\left (a+2\,\ln \left (x\right )\right )}^2\,{\left (e\,x\right )}^m \,d x \] Input:

int(tanh(a + 2*log(x))^2*(e*x)^m,x)
 

Output:

int(tanh(a + 2*log(x))^2*(e*x)^m, x)
 

Reduce [F]

\[ \int (e x)^m \tanh ^2(a+2 \log (x)) \, dx=\frac {e^{m} \left (x^{m} e^{2 a} m \,x^{5}-3 x^{m} e^{2 a} x^{5}+4 e^{2 a} \left (\int \frac {x^{m}}{e^{4 a} m \,x^{8}-3 e^{4 a} x^{8}+2 e^{2 a} m \,x^{4}-6 e^{2 a} x^{4}+m -3}d x \right ) m^{3} x^{4}-4 e^{2 a} \left (\int \frac {x^{m}}{e^{4 a} m \,x^{8}-3 e^{4 a} x^{8}+2 e^{2 a} m \,x^{4}-6 e^{2 a} x^{4}+m -3}d x \right ) m^{2} x^{4}-20 e^{2 a} \left (\int \frac {x^{m}}{e^{4 a} m \,x^{8}-3 e^{4 a} x^{8}+2 e^{2 a} m \,x^{4}-6 e^{2 a} x^{4}+m -3}d x \right ) m \,x^{4}-12 e^{2 a} \left (\int \frac {x^{m}}{e^{4 a} m \,x^{8}-3 e^{4 a} x^{8}+2 e^{2 a} m \,x^{4}-6 e^{2 a} x^{4}+m -3}d x \right ) x^{4}-3 x^{m} m x -7 x^{m} x +4 \left (\int \frac {x^{m}}{e^{4 a} m \,x^{8}-3 e^{4 a} x^{8}+2 e^{2 a} m \,x^{4}-6 e^{2 a} x^{4}+m -3}d x \right ) m^{3}-4 \left (\int \frac {x^{m}}{e^{4 a} m \,x^{8}-3 e^{4 a} x^{8}+2 e^{2 a} m \,x^{4}-6 e^{2 a} x^{4}+m -3}d x \right ) m^{2}-20 \left (\int \frac {x^{m}}{e^{4 a} m \,x^{8}-3 e^{4 a} x^{8}+2 e^{2 a} m \,x^{4}-6 e^{2 a} x^{4}+m -3}d x \right ) m -12 \left (\int \frac {x^{m}}{e^{4 a} m \,x^{8}-3 e^{4 a} x^{8}+2 e^{2 a} m \,x^{4}-6 e^{2 a} x^{4}+m -3}d x \right )\right )}{e^{2 a} m^{2} x^{4}-2 e^{2 a} m \,x^{4}-3 e^{2 a} x^{4}+m^{2}-2 m -3} \] Input:

int((e*x)^m*tanh(a+2*log(x))^2,x)
 

Output:

(e**m*(x**m*e**(2*a)*m*x**5 - 3*x**m*e**(2*a)*x**5 + 4*e**(2*a)*int(x**m/( 
e**(4*a)*m*x**8 - 3*e**(4*a)*x**8 + 2*e**(2*a)*m*x**4 - 6*e**(2*a)*x**4 + 
m - 3),x)*m**3*x**4 - 4*e**(2*a)*int(x**m/(e**(4*a)*m*x**8 - 3*e**(4*a)*x* 
*8 + 2*e**(2*a)*m*x**4 - 6*e**(2*a)*x**4 + m - 3),x)*m**2*x**4 - 20*e**(2* 
a)*int(x**m/(e**(4*a)*m*x**8 - 3*e**(4*a)*x**8 + 2*e**(2*a)*m*x**4 - 6*e** 
(2*a)*x**4 + m - 3),x)*m*x**4 - 12*e**(2*a)*int(x**m/(e**(4*a)*m*x**8 - 3* 
e**(4*a)*x**8 + 2*e**(2*a)*m*x**4 - 6*e**(2*a)*x**4 + m - 3),x)*x**4 - 3*x 
**m*m*x - 7*x**m*x + 4*int(x**m/(e**(4*a)*m*x**8 - 3*e**(4*a)*x**8 + 2*e** 
(2*a)*m*x**4 - 6*e**(2*a)*x**4 + m - 3),x)*m**3 - 4*int(x**m/(e**(4*a)*m*x 
**8 - 3*e**(4*a)*x**8 + 2*e**(2*a)*m*x**4 - 6*e**(2*a)*x**4 + m - 3),x)*m* 
*2 - 20*int(x**m/(e**(4*a)*m*x**8 - 3*e**(4*a)*x**8 + 2*e**(2*a)*m*x**4 - 
6*e**(2*a)*x**4 + m - 3),x)*m - 12*int(x**m/(e**(4*a)*m*x**8 - 3*e**(4*a)* 
x**8 + 2*e**(2*a)*m*x**4 - 6*e**(2*a)*x**4 + m - 3),x)))/(e**(2*a)*m**2*x* 
*4 - 2*e**(2*a)*m*x**4 - 3*e**(2*a)*x**4 + m**2 - 2*m - 3)